sb200 lecture 5 30 september 2008 jeremy gunawardenamax delbruck nobel in physiology 1969 christian...

A systems approach to biology

SB200

Lecture 530 September 2008

Jeremy Gunawardena

jeremy@hms.harvard.edu

Recap of Lecture 4

matrix exponential

exp(A) = 1 + A + A2/2 + ... + Ak/k! + ...

matrices as linear transformations

similarity

B = T-1AT

dx/dt = Ax

x(t) = exp(At)x0

linear transformations

> 0

< 0

= 0

normal forms

complex case disc < 0 complex eigenvalues

stable spiral

Tr = -3 det = 7 disc = -19eigenvalues = -1.5 2.18 i

Complex eigenvalues imply (damped) oscillation, with frequency given by the imaginary part of the eigenvalue

x1

x2

eigenvalues = a i b

trace

dete

rmin

ant

= 0

det > 0 , Tr > 0

> 0 > 0

saddles

spirals

nodes

sinks sources

cent

ers

< 0 , Tr < 0

det > 0 , Tr < 0

<

0 , T

r =

0

det < 0

> 0 , Tr < 0

awkward case 1 disc = 0

non-generic (degenerate) case

cant make up its mind whether to be a node or a spiral

center

awkward case 2 disc < 0, Tr A = 0

Tr = 0 det = 6 disc = -24eigenvalues = 2.45 i

cant make up its mind whether to be stable or unstable

robust oscillations require nonlinearity

x1

x2

stable

x1

stable

unstable

x2

BACK TO PHAGE LAMBDA

does this model capture the biology?

repressor high/on - lysogeny

increased degradation of

repressor off - lysis switch is sluggish, not sharp

repressor high/on - lysogeny

off state is unstable

how could the design be changed

to make the switch sharper

and/or

the off-state stable?

but re-design f or g to bend the nullcline(s)

use the same basic design as before

creating two stable nodesseparated by an unstable saddle

a sigmoidal dose-response curve

saturating

derivative decreasing

derivative increasing

sigmoidal = S-shaped

sigmoidal curves correspond to

unimodal probability distributions

correspond to

differentiation

area under the curve(integration)

1

0

0.9

0.1

sigmoidal curves have two independent features

they create a threshold

normalised 1

00.1

normalised

they switch from low to high

x0.1 x0.1 x0.9

good switch = low (x0.9 - x0.1)good threshold = high x0.1

good threshold, poor switch good switch, poor threshold

different measures of “ switching-ness”

1

0

0.9

0.1

normalised

x0.1 x0.9

x0.9 / x0.1 - COOPERATIVITY INDEX

CI = 81 for the standard hyperbolic curve

a = 1; h = 2, 6, 20

HILL COEFFICIENT

CI = 811/h

Johan will tell you about another measure in his lectures

ultrasensitivity - a small change in dose causes a large change in response Goldbeter & Koshland, PNAS 78:6840-4 1981

ultrasensitive - if CI < 81

cooperativity

one interaction (eg: a binding event) changes the effect ofa subsequent interaction

an important mechanism for creating sigmoidal dose-responses

haemoglobin

partial pressure of oxygen

% o

xyge

n sa

tura

tion

cooperativity in oxygen binding to haemoglobin

Christian Bohr, Boris Hasselbach & August Krogh, Skand. Arch. Physiol., 16:401-12, 1904

Haemoglobin cooperativity is based on allostery

August KroghNobel in Physiology 1920

Niels BohrNobel in Physics1922

Max DelbruckNobel in Physiology1969

Christian Bohr

Father/son

Teacher/student

Teacher/student

phage lambda creates cooperativity through PROMOTER STRUCTURE

haemoglobin creates cooperativity throughALLOSTERY

gene expression depends on promoter structure

RNA polymerase

gene coding region transcriptionstart site

mRNA transcript

RNA polymerase tra

nscr

iptio

n fa

ctor

transcription

factor

trans

crip

tion

fact

or

transcription factor binding sites

promoter region

promoter region

amino acids

primary sequence

secondary structure – helices and sheets

A, C, D, E, F, G, H, I, K, L, M, N, P, Q, R, S, T, V, W, Y AAAAAAGAGP EMVRGQVFDV GPRYTNLSYI GEGAYGMVCS AYDNVNKVRV AIKKISPFEH QTYCQRTLRE IKILLRFRHE NIIGINDIIR APTIEQMKDV YIVQDLMETD LYKLLKTQHL SNDHICYFLY QILRGLKYIH SANVLHRDLK PSNLLLNTTC DLKICDFGLA RVADPDHDHT GFLTEYVATR WYRAPEIMLN SKGYTKSIDI WSVGCILAEM LSNRPIFPGK HYLDQLNHIL GILGSPSQED LNCIINLKAR NYLLSLPHKN KVPWNRLFPN ADSKALDLLD KMLTFNPHKR IEVEQALAHP YLEQYYDPSD EPIAEAPFKF DMELDDLPKE KLKELIFEET ARFQPGYRS

http://www.rcsb.org/pdb/

Erk2 – Extracellular signal Regulated Kinase SwissProt P28482

proteins

PDB 1ERK

tertiary structure – do mains

proteins

Tony Pawson's lab http://pawsonlab.mshri.on.ca/

lambda repressor – dimerisation and DNA binding

dimerisation

DNA binding

1lmb.pdb Beamer & Pabo J Mol Biol 227:177, 1992

lambda repressor – binding to operator region

DNA binding

OR1OR2OR3

cooperative binding of repressor dimer

to OR1 and OR2

repressor transcribed at Iow basal rate

~11x increase in repressor transcription

repressor transcription turned off

calculating the rate of repressor expression

OR1OR2OR3

unoccupied

occupied by repressor dimer

D1

D2

D3

D0

Shea-Ackers modelAckers, Johnson & Shea, PNAS 79:1129-33 1982

a general statistical mechanical model for transcription factor binding

simplify

OR1OR2OR3

calculate the probabilities of finding repressor bound to DNA in each state (D0, D1, D2, D3)

calculate the rate of gene transcription as an average over this probability distribution